I taught handstands for several years in group and one-on-one formats.
Regarding handstands and "strength:" Once you have the basic wrist strength/mobility and requisite shoulder mobility, handstanding is about exerting the least amount of effort to stay up. I have trained people who could hardly do a clean set of pushups but could do handstands just fine. There are handstand skills (press handstand varieties, bent arm handstand work such as HSPUs) that require and build strength, but a basic handstand, unless done for considerably long hold-times, is not a strength skill imo. It's worth noting that many very strong calisthenics athletes cannot hold a freestanding handstand for longer than a minute.
Regarding carry-over to lifting things that are not your own body, and whether or not it is a "necessary" athletic skill to develop: Carryover to anyting not handstand or calisthenics oriented is highly debatable. I've seen calisthenics athletes who can do very high reps of HSPU do barbell OVHP significantly greater than bodyweight, and I've also seen the exact opposite. I wouldn't dare draw a line in the sand one way or the other.
Bottom line, and my two cents: if you're interested, and you have fun in the process, do them. That said, acheiving a solid freestanding handstand can take anywhere from 6 months to 10 years, depending on your physical condition when you start training them, and how dedicated you are to training them. I'll be honest, even with decent shoulder mobility at the start, you're considering potentially years of staring at the floor trying not to fall over, and that can be just for a basic handstand. If you're up for something that requires discipline, awesome. If you don't think you can or want to spend hours each week doing shoulder prehab and very basic handstand drills, it might not be for you.
Personally, I still want to get back to them after a few years of setbacks (for a variety of reasons). It feels really cool to be able to balance and move around upside down. I think it's fun, personally. It's just not something you can hurry your way through.